Integrability for solutions to quasilinear elliptic systems

Integrability for solutions to quasilinear elliptic systems

Francesco Leonetti, Pier Vincenzo Petricca

Abstract. In this paper we prove an estimate for the measure of superlevel sets of weak solutions to quasilinear elliptic systems in divergence form. In some special cases, such an estimate allows us to improve on the integrability of the solution.

Keywords: level set, integrability, solution, quasilinear, elliptic, system Classification: 35J62, 35J47, 35D10

1. Introduction

We deal with regularity properties for weak solutions u : Ω ⊂ Rⁿ → R^N of quasilinear systems in divergence form

(1.1) −

n

X

i=1

Di





n

X

j=1 N

X

β=1

a^αβ_ij (x, u(x))Dju^β(x)



= 0, x∈Ω, α= 1, . . . , N.

The coefficientsa^αβ_ij (x, u) are only measurable with respect tox; they are contin- uous with respect tou; moreover, they are bounded and elliptic. We assume that the solutionuof (1.1) is bounded on∂Ω:

(1.2) u∈L^∞(∂Ω).

De Giorgi’s counterexample [1] shows that, in general, boundedness on∂Ω does not imply boundedness inside Ω for weak solutionsuof elliptic systems (1.1); see also [8]. In order to get boundedness inside Ω, we need additional assumptions on the coefficients. Ifa^αβ_ij (x, u) are diagonal

(1.3) a^γβ_ij (x, u) = 0 forβ 6=γ

then theNequations (1.1) are decoupled and maximum principle applies to every componentu^γ ofu= (u¹, . . . , u^N):

(1.4) sup

Ω

u^γ ≤sup

∂Ω

u^γ.

We acknowledge the support of MIUR.

In [7] the authors assume that coefficients are diagonal only for large values ofu^γ: (1.5) θ^γ≤u^γ =⇒ a^γβ_ij (x, u) = 0 forβ6=γ;

then it results that

(1.6) sup

Ω

u^γ≤max

θ^γ; sup

∂Ω

u^γ

,

see also [6] and [5]. In this paper we no longer assume that off-diagonal coefficients vanish; we only know that they are small when u^γ is large: there exists q > 0 such that

(1.7) 0< θ^γ ≤ |u^γ| =⇒ |a^γβ_ij (x, u)| ≤ c1

|u^γ|^q for β6=γ;

then we are able to estimate the measure of superlevels as follows

(1.8) |{|u^γ|> s}| ≤ c2

s^2∗(1+q)

for every s > 0; note that 2^∗ = 2n/(n−2) is the Sobolev exponent. Such an inequality is a special case of a general estimate that we prove for every system (1.1): in the special case (1.7) we get (1.8). Thenu^γ turns out to be in the weak Lebesgue (or Marcinkiewicz) space with exponent 2^∗(1 +q):

(1.9) u^γ ∈L²_weak^∗(1+q)(Ω).

Note that weak solutions u of system (1.1) are taken from the Sobolev space W^1,2(Ω;R^N); the embedding guarantees that the integrability of u reaches 2^∗: our result (1.9) improves such integrability, since 2^∗ <2^∗(1 +q). In Section 2 we collect precise assumptions and results; Section 3 is devoted to the proof.

We end this introduction by recalling that [4] deals with the linear case and the author provesL^∞bounds under an assumption on the dispersion of eigenvalues.

Eventually, we thank the referee for valuable remarks.

2. Assumptions and results

Let Ω be a bounded open subset ofRⁿ,n≥3. ForN≥2, leta^αβ_ij : Ω×R^N →R be Carath´eodory functions, that is, a^αβ_ij (x, y) are measurable with respect to x and continuous with respect toy. We assume that coefficients are bounded: there existsc3∈(0,+∞) such that

(2.1) |a^αβ_ij (x, y)| ≤c3

for almost every x ∈ Ω, for every y ∈ R^N, for all i, j ∈ {1, . . . , n}, for any α, β∈ {1, . . . , N}. Letν∈(0,+∞); we assume ellipticity of diagonal coefficients

a^γγ_ij for large values of the corresponding component ofy: for everyγ∈ {1, . . . , N} there existsθ^γ ∈(0,+∞) such that

(2.2) θ^γ ≤ |y^γ| =⇒ ν|ξ|²≤

n

X

i,j=1

a^γγ_ij(x, y)ξjξi

for almost every x ∈ Ω, for any ξ ∈ Rⁿ. In order to deal with off-diagonal coefficients a^γβ_ij we need to introduce the following supremum: for every γ ∈ {1, . . . , N}, for everyL∈(0,+∞) we define

(2.3) g^γ(L) = max

i,j max

β6=γ sup

|y^γ|>L

sup

x |a^γβ_ij (x, y)|,

where maxi,j is taken over all i, j ∈ {1, . . . , n}; maxβ6=γ is taken over all β ∈ {1, . . . , N} \ {γ}; sup_|y^γ_|>L is taken over all y ∈R^N with |y^γ|> L; sup_x is the essential supremum taken over almost every x ∈ Ω. Note that L → g^γ(L) is decreasing; moreover, assumption (2.1) guarantees that 0≤g^γ(L)≤c3.

We prove the following

Theorem 2.1. Under the previous assumptions(2.1),(2.2), letu= (u¹, . . . , u^N) be a weak solution of the system(1.1), that is,u∈W^1,2(Ω,R^N)and

(2.4) Z

Ω N

X

α,β=1 n

X

i,j=1

a^αβ_ij (x, u(x))Dju^β(x)Div^α(x)dx= 0 ∀v∈W₀^1,2(Ω,R^N).

Then every componentu^γ of u= (u¹, . . . , u^N)satisfies (2.5) |{x∈Ω :|u^γ(x)|>2L}| ≤c4

g^γ(L) L

2^∗

, where

(2.6) c4= 2

2(n−1)(N−1)n²

(n−2)ν ||Du||L²(Ω)

^2∗ ,

|E|is the Lebesgue measure ofE⊂Rⁿ and2^∗= 2n/(n−2)is the Sobolev expo- nent. Inequality (2.5)holds true for everyL≥max{θ^γ; sup_∂Ωu^γ;−inf∂Ωu^γ}.

Remark 2.1. With no extra assumption, g^γ(L)≤c3 and decay (2.5) does not improve on Sobolev embeddingW^1,2⊂L^2∗.

Remark 2.2. When off-diagonal coefficients a^γβ_ij vanish for large values of|y^γ| (2.7) 0< θ^γ ≤ |y^γ| =⇒ a^γβ_ij (x, y) = 0 for β6=γ,

then g^γ(L) = 0 for L ≥ θ^γ and decay (2.5) says that some superlevel has zero measure, thus we haveL^∞estimates: this is already known since [7] and [6].

Remark 2.3. Now we assume that off-diagonal coefficientsa^γβ_ij (x, y) do not van- ish any more, but they are small when the corresponding componenty^γ is large:

there existq, c5∈(0,+∞) such that

(2.8) 0< θ^γ ≤ |y^γ| =⇒ |a^γβ_ij (x, y)| ≤ c5

|y^γ|^q for β6=γ;

theng^γ(L)≤c5/L^q forL≥θ^γ and (2.5) gives us

(2.9) |{|u^γ|>2L}| ≤ c6

L^2∗(1+q)

for everyL≥max{θ^γ; sup_∂Ωu^γ;−inf∂Ωu^γ}. This allows us to improve the inte- grability ofuas follows:

Theorem 2.2. Under the previous assumptions(2.1),(2.2), letu= (u¹, . . . , u^N) be a weak solution of the system(1.1), that is, u∈W^1,2(Ω,R^N)and (2.4)holds true. In addition, we assume that off-diagonal coefficients satisfy(2.8). Moreover, we require that

(2.10) −∞<inf

∂Ωu^γ and sup

∂Ω

u^γ <+∞, for everyγ= 1, . . . , N; thenuattains higher integrability:

(2.11) u∈L²weak^∗(1+q)(Ω;R^N).

Remark 2.4. Please, note that assumption (2.10) implies

max{θ^γ; sup_∂Ωu^γ;−inf∂Ωu^γ} < +∞, thus we can use (2.9): such an estimate and boundedness of Ω guarantee (2.11). Since we aim at higher integrability of u, it would be nice to have the same result only assuming enough integrability of the boundary datum, instead of requiring boundedness on the boundary of Ω as in (2.10). It would also be interesting to have a local version of the previous Theorem without any restriction on the boundary datum.

Remark 2.5. Note that reverse H¨older inequality gives us higher integrability of the gradient: Du ∈ L^2+ǫ, see [2] and Chapter 6 in [3]; this improves on the integrability ofuby means of Sobolev embedding: u∈L^(2+ǫ)∗. In order to have global higher integrability, both the boundary of Ω and the boundary datum have to be regular enough, see Theorem 6.8 at page 209 in [3]. Moreover, it seems that strong ellipticity ofa^αβ_ij is required; in the present paper we need ellipticity only for diagonal entriesa^γγ_ij and only for large values ofu^γ. Please, note that we do not assume ellipticity for small values ofu^γ: on such a set u^γ is bounded but it might oscillate very much and the gradientDu^γ might lose regularity.

3. Proof of Theorem 2.1

We fixγ ∈ {1, . . . , N} and we take L ∈R withL ≥max{θ^γ; sup∂Ωu^γ}> 0, where u^γ is the γ-th component of u = (u¹, . . . , u^N). Since L ≥ sup∂Ωu^γ, we

have max{u^γ−L; 0} ∈W₀^1,2(Ω). We define v= (v¹, . . . , v^N) as follows (3.1)

(v^α= 0 if α6=γ v^γ = max{u^γ−L; 0} otherwise.

Note that (3.2)

(Dv^α= 0 if α6=γ Dv^γ = 1{u^γ>L}Du^γ otherwise

where 1E is the characteristic function of the set E, that is, 1E(x) = 1 ifx∈E and 1E(x) = 0 ifx /∈E. We insert such a test functionv into (2.4):

(3.3)

0 = Z

Ω n

X

i,j=1 N

X

α,β=1

a^αβ_ij (x, u(x))Dju^β(x)Div^α(x)dx

= Z

Ω n

X

i,j=1 N

X

β=1

a^γβ_ij (x, u(x))Dju^β(x)1{u^γ>L}(x)Diu^γ(x)dx

= Z

{u^γ>L}

n

X

i,j=1

a^γγ_ij (x, u(x))Dju^γ(x)Diu^γ(x)dx

+ Z

{u^γ>L}

n

X

i,j=1

X

β6=γ

a^γβ_ij (x, u(x))Dju^β(x)Diu^γ(x)dx.

Then

(3.4)

Z

{u^γ>L}

n

X

i,j=1

a^γγ_ij (x, u(x))Dju^γ(x)Diu^γ(x)dx

=− Z

{u^γ>L}

n

X

i,j=1

X

β6=γ

a^γβ_ij (x, u(x))Dju^β(x)Diu^γ(x)dx.

SinceL≥θ^γ, we can use ellipticity (2.2) and we get (3.5) ν

Z

{u^γ>L}

|Du^γ|²dx≤ Z

{u^γ>L}

n

X

i,j=1

a^γγ_ij (x, u(x))Dju^γ(x)Diu^γ(x)dx.

We keep in mind the definition (2.3) forg^γ(L) and we have

(3.6)

− Z

{u^γ>L}

n

X

i,j=1

X

β6=γ

a^γβ_ij (x, u(x))Dju^β(x)Diu^γ(x)dx

≤n²(N−1)g^γ(L) Z

{u^γ>L}

|Du||Du^γ|dx.

Equality (3.4) and estimates (3.5), (3.6) merge into (3.7) ν

Z

{u^γ>L}

|Du^γ|²dx≤n²(N−1)g^γ(L) Z

{u^γ>L}

|Du||Du^γ|dx.

We use H¨older inequality on the right hand side in order to get

(3.8) ν

Z

{u^γ>L}

|Du^γ|²dx

≤n²(N−1)g^γ(L) Z

{u^γ>L}

|Du|²dx

!1/2

Z

{u^γ>L}

|Du^γ|²dx

!1/2

. We divide both sides by (R

{u^γ>L}|Du^γ|²dx)^1/2 and we get (3.9)

Z

{u^γ>L}

|Du^γ|²dx

!1/2

≤n²(N−1)g^γ(L) ν

Z

{u^γ>L}

|Du|²dx

!1/2

. We keep in mind thatv^γ = max{u^γ−L; 0} ∈W₀^1,2(Ω) andn≥3, thus Sobolev inequality and (3.9) allow us to write

(3.10) Z

{u^γ>L}

(u^γ−L)^2∗dx=||v^γ||²_L^∗2∗(Ω)≤

2(n−1)

n−2 ||Dv^γ||L²(Ω)

2^∗

=





2(n−1) n−2

Z

{u^γ>L}

|Du^γ|²dx

!1/2



2^∗

≤

2(n−1) n−2

n²(N−1)

ν [g^γ(L)]||Du||_L2(Ω)

^2∗ . SinceL >0, it turns out that{u^γ>2L} ⊂ {u^γ > L}, thus

(3.11)

L^2∗|{u^γ >2L}|= Z

{u^γ>2L}

(2L−L)^2∗dx

≤ Z

{u^γ>2L}

(u^γ−L)^2∗dx≤ Z

{u^γ>L}

(u^γ−L)^2∗dx.

Inequalities (3.10) and (3.11) merge into (3.12) |{u^γ>2L}| ≤

2(n−1)n²(N−1)

(n−2)ν ||Du||L²(Ω)

[g^γ(L)]

L ^2∗

.

This estimate holds true for everyL≥max{θ^γ; sup_∂Ωu^γ}>0. Since−inf∂Ωu^γ = sup∂Ω(−u^γ), ifL≥max{θ^γ;−inf∂Ωu^γ}>0, then we can apply the previous in- equality (3.12) to−u. This ends the proof of Theorem 2.1.

References

[1] De Giorgi E.,Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital.4(1968), 135–137.

[2] Giaquinta M., Modica G., Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Angew. Math.311/312(1979), 145–169.

[3] Giusti E.,Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003.

[4] Leonardi S., A maximum principle for linear elliptic systems with discontinuous coeffi- cients, Comment. Math. Univ. Carolin.45(2004), 457–474.

[5] Leonetti F., Petricca P.V., Regularity for solutions to some nonlinear elliptic systems, Complex Var. Elliptic Equ., to appear.

[6] Mandras F.,Principio di massimo per una classe di sistemi ellittici degeneri quasi lineari, Rend. Sem. Fac. Sci. Univ. Cagliari46(1976), 81–88.

[7] Neˇcas J., Star´a J.,Principio di massimo per i sistemi ellittici quasi-lineari non diagonali, Boll. Un. Mat. Ital.6(1972), 1–10.

[8] ˇSver´ak V., Yan X., Non-Lipschitz minimizers of smooth uniformly convex functionals, Proc. Natl. Acad. Sci. USA99(2002), 15269–15276.

Dipartimento di Matematica Pura ed Applicata, Universit`a di L’Aquila, 67100 L’Aquila, Italy

E-mail: [email protected]

Via Sant’Amasio 18, 03039 Sora, Italy

(Received January 29, 2010, revised May 10, 2010)